Question: Point $O$ is the center of an ellipse with major axis $\overline{AB}$ and minor axis $\overline{CD}.$  Point $F$ is one focus of the ellipse.  If $OF = 6$ and the diameter of the inscribed circle of triangle $OCF$ is 2, compute the product $(AB)(CD).$
Solution: Let $a = OA = OB$ and $b = OC = OD.$  Then $a^2 - b^2 = OF^2 = 36.$

[asy]
unitsize(0.5 cm);

path ell = xscale(5)*yscale(3)*Circle((0,0),1);
pair A, B, C, D, F, O;

A = (5,0);
B = (-5,0);
C = (0,3);
D = (0,-3);
F = (4,0);
O = (0,0);

draw(ell);
draw(A--B);
draw(C--D);
draw(C--F);
draw(incircle(O,C,F));

label("$A$", A, E);
label("$B$", B, W);
label("$C$", C, N);
label("$D$", D, S);
label("$F$", F, S);
label("$O$", O, SW);
[/asy]

In general, if a right triangle has legs $x$ and $y,$ and hypotenuse $z,$ then its inradius is given by
\[\frac{x + y - z}{2}.\]Hence, the diameter of the incircle of triangle $OCF$ is
\[OC + OF - CF = 2.\]Then $b + 6 - a = 2,$ so $a - b = 4.$

By difference of squares on the equation $a^2 - b^2 = 36,$ $(a + b)(a - b) = 36,$ so
\[a + b = \frac{36}{a - b} = 9.\]With the equation $a - b = 4,$ we can solve to get $a = \frac{13}{2}$ and $b = \frac{5}{2}.$

Then $AB = 13$ and $CD = 5,$ so $(AB)(CD) = \boxed{65}.$